The present invention relates to a surface acoustic wave resonator which is compact in size and has a high Q and a low resonance resistance.
In general, surface acoustic wave resonators (to be referred to as "SAW resonators" in this specification) comprises a piezoelectric substrate, a pair of surface acoustic wave grating reflectors formed on the substrate and an interdigital transducer formed on the substrate between the reflectors. Surface acoustic waves propagate between the reflector pair and resonate at a certain frequency, and this resonance is coupled to an electric circuit via terminals. Such SAW resonators as described above have been known as "the cavity type". It has been well known that in the prior art cavity type SAW resonators, their Q and resonance resistance R.sub.1 are dependent upon a maximum absolute value .vertline..GAMMA..vertline..sub.max of the reflection coefficient .GAMMA. of a reflector and the radiation conductance G.sub.a of a transducer and have the following relations: EQU Q.varies.1/(1-.vertline..GAMMA..vertline..sub.max.sup.2) (1)
and ##EQU2## It follows, therefore, that in order to obtain a SAW resonator with a high Q and a low R.sub.1, the reflection coefficient .vertline..GAMMA..vertline..sub.max be as close to unity as possible and the radiation conductance G.sub.a be increased as high as possible. As a result, both the reflectors and transducer need an extremely large number of electrodes. When a substrate consists of a quartz crystal having a lesser degree of surface acoustic wave reflecting capability through its piezoelectricity, a number of 500 to 1,000 reflector electrodes are arranged in general so that it has been difficult to make the SAW resonators compact in size.
In order to eliminate this drawback, there has been proposed a method for improving the reflection coefficient by forming periodic arrays of grooves on the surface of the substrate.
The prior art cavity type SAW resonators are so designed that the resonance is obtained at a frequency f.sub.R (to be referred to as "the center frequency of the reflector" in this specification) at which the reflection coefficient .vertline..GAMMA..vertline. of the reflector becomes maximum. The resonance is also affected by the spacing between the transducer and the reflector, thus the spacing being one of the most important design criteria. It has been also well known in the art that an optimum spacing is obtained from the following equation: EQU l.sub.1 +l.sub.2 =(n/2+1/4).lambda. (3)
where
l.sub.1 is the spacing between the transducer and one of the two reflectors; PA1 l.sub.2 is the spacing between the transducer and the other reflector; PA1 n: a positive integer; and PA1 .lambda.: the wavelength of surface acoustic waves at the resonance frequency. PA1 where PA1 C.sub.2T : the frequency decrease due to the periodic structure of the transducer; PA1 C.sub.2R : the frequency decrease due to the periodic structure of the reflector; and PA1 q.sub.T =.pi.C.sub.1T N PA1 where N is the number of electrode finger pairs in the transducer.
That is, in the design of the prior art SAW resonators the spacings between the transducer and the reflectors are obtained from Eq. (3) so that the resonance is obtained at the center frequency f.sub.R of the reflector.
So far in the design of the SAW resonators, the characteristics of the reflectors have attracted much more attention than those of the transducer and have been investigated in detail. However, there has not been disclosed any technical report and appear particularly concerning the characteristics of the transducer. Therefore, the inventors made extensive studies and experiments of the frequency dependency of the reflection coefficient of the reflectors and the acoustic radiation conductance of the transducer in the prior art SAW resonator in which the transducer electrode period is equal to the reflector electrode period. The results showed that the reflector center frequency f.sub.R is spaced apart from a frequency f.sub.T (to be referred to as "the center frequency of the transducer") at which the acoustic radiation conductance becomes maximum. That is, f.sub.T &lt;f.sub.R. Furthermore, it was found out that the acoustic radiation conductance G.sub.a decreases considerably as compared with its maximum value at frequencies in the vicinity of the center frequency f.sub.R of the reflector.
These observed facts show that in the prior art SAW resonators, the frequency response of the transducer is not fully utilized. As a consequence and as it is readily seen from Eq. (2), in order to lower the resonance resistance R.sub.1, the decrease in the radiation conductance G.sub.a at the resonance frequency must be compensated for by an increase in the reflection coefficient. As a result, the reflectors must consist of a large number of electrodes.
Meanwhile it has been proposed to provide the reflector with a periodic array of grooves so that the reflection coefficient may be increased. However, the frequency dependency of the acoustic conductance of the transducer is still not fully utilized so that the number of reflector electrodes cannot be reduced to a desired degree. As a result, the prior art SAW resonators cannot be made compact in size.